Physica D 237 (2008) 1937–1944www.elsevier.com/locate/physd

Blowup or no blowup? The interplay between theory and numerics

Thomas Y. Houa,∗, Ruo Lib

a Applied and Computational Math, 217-50, Caltech, Pasadena, CA 91125, USAb LMAM&School of Mathematical Sciences, Peking University, Beijing 100871, China

Available online 25 January 2008

Abstract

The question of whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data has been anoutstanding open problem in fluid dynamics and mathematics. Recent studies indicate that the local geometric regularity of vortex lines can leadto dynamic depletion of vortex stretching. Guided by the local non-blowup theory, we have performed large scale computations of the 3D Eulerequations on some of the most promising blowup candidates. Our results show that there is tremendous dynamic depletion of vortex stretching. Thelocal geometric regularity of vortex lines and the anisotropic solution structure play an important role in depleting the nonlinearity dynamicallyand thus prevents a finite time blowup.c© 2008 Elsevier B.V. All rights reserved.

PACS: 47.32.C-; 47.11.Kb

Keywords: Finite time singularities; 3D Euler equations; Spectral methods

1. Introduction

The question of whether the 3D incompressible Eulerequations can develop a finite time singularity from smoothinitial data is one of the most outstanding open problemsin fluid dynamics and mathematics. This open problem isclosely related to the Clay Millennium Open Problem onthe 3D Navier–Stokes equations. The understanding of thisproblem could improve our understanding on the onset ofturbulence and the intermittency properties of turbulent flows.A main difficulty in answering this question is the presence ofvortex stretching, which gives a formal quadratic nonlinearityin vorticity. There have been many computational efforts insearching for finite time singularities of the 3D Euler equations,see e.g. [2–5,11–13,17,18,21–23]. For a more comprehensivereview of this subject, we refer the reader to the book by Majdaand Bertozzi [20] and the excellent review article by J. Gibbonin this issue [10].

Computing Euler singularities numerically is an extremelychallenging task. First of all, it requires huge computational

∗ Corresponding author. Tel.: +1 626 395 4546; fax: +1 626 578 0124.E-mail address: hou@acm.caltech.edu (T.Y. Hou).

0167-2789/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2008.01.018

resources. Tremendous resolutions are required to capture thenearly singular behavior of the Euler equations. Secondly, onehas to perform a careful convergence study. It is dangerous tointerpret the blowup of an under-resolved computation as anevidence of finite time singularities for the 3D Euler equations.Thirdly, if we believe that the numerical solution we computeleads to a finite time blowup, we need to demonstrate thevalidation of the asymptotic blowup rate, i.e. is the blowuprate ‖ω‖L∞ ≈

C(T −t)α asymptotically valid as t → T ?

One also needs to check if the blowup rate of the numericalsolution is consistent with the Beale–Kato–Majda non-blowupcriterion [1] and other non-blowup criteria [7–9]. The interplaybetween theory and numerics is clearly essential in our searchfor Euler singularities.

There has been some interesting development in thetheoretical understanding of the 3D incompressible Eulerequations. It has been shown that the local geometric regularityof vortex lines can play an important role in depletingnonlinear vortex stretching [6–9]. In particular, the recentresults obtained by Deng, Hou, and Yu [8,9] show thatgeometric regularity of vortex lines, even in an extremelylocalized region containing the maximum vorticity, can leadto depletion of nonlinear vortex stretching, thus avoiding finite

1938 T.Y. Hou, R. Li / Physica D 237 (2008) 1937–1944

time singularity formation of the 3D Euler equations. To obtainthese results, Deng–Hou–Yu [8,9] explore the connectionbetween the stretching of local vortex lines and the growthof vorticity. In particular, they show that if the vortex linesnear the region of maximum vorticity satisfy some localgeometric regularity conditions and the maximum velocityfield is integrable in time, then no finite time blowup ispossible. These localized non-blowup criteria provide strongerconstraints on the local geometry of a potential finite timesingularity. They can be used to re-examine some of the well-known numerical evidences for finite time singularities of the3D Euler equations.

2. A brief review

We begin with a brief review on the subject. Due to theformal quadratic nonlinearity in vortex stretching, only shorttime existence is known for the 3D Euler equations. One ofthe most well-known results on the 3D Euler equations is dueto Beale–Kato–Majda [1] who show that the solution of the 3DEuler equations blows up at T ∗ if and only if

∫ T ∗

0 ‖ω‖∞(t) dt =

∞, where ω is vorticity.There have been some interesting recent theoretical

developments. In particular, Constantin–Fefferman–Majda [7]show that local geometric regularity of the unit vorticity vectorcan lead to depletion of the vortex stretching. Let ξ = ω/|ω|

be the unit vorticity vector and u be the velocity field. Roughlyspeaking, Constantin–Fefferman–Majda show that if (1) ‖u‖∞

is bounded in a O(1) region containing the maximum vorticity.(2)

∫ t0 ‖∇ξ‖

2∞dτ is uniformly bounded for t < T , then the

solution of the 3D Euler equations remains regular up to t = T .There have been some numerical evidences which suggest a

finite time blowup of the 3D Euler equations. One of the mostwell-known examples is the finite time collapse of two anti-parallel vortex tubes by Kerr [17,18]. In Kerr’s computations,he used a pseudo-spectral discretization in the x and ydirections, and a Chebyshev discretization in the z directionwith resolution of order 512 × 256 × 192. His computationsshowed that the maximum vorticity blows up like O((T − t)−1)

with T = 18.9. In his subsequent paper [18], Kerr showed thatthe maximum velocity blows up like O((T − t)−1/2) with Tbeing revised to T = 18.7. It is worth noting that there is stilla considerable gap between the predicted singularity time T =

18.7 and the final time t = 17 of Kerr’s computations which heused as the primary evidence for the finite time singularity.

Kerr’s blowup scenario is consistent with the Beale–Kato–Majda non-blowup criterion [1] and the Constantin–Fefferman–Majda non-blowup criterion [7]. But it falls into the critical caseof the Deng–Hou–Yu local non-blowup criteria [8,9]. Below wedescribe the local non-blowup criteria of Deng–Hou–Yu.

3. The local non-blowup criteria of Deng–Hou–Yu [8,9]

Motivated by the result of [7], Deng, Hou, and Yu [8] haveobtained a sharper non-blowup condition which uses only verylocalized information of the vortex lines. Assume that at eachtime t there exists some vortex line segment L t on which the

local maximum vorticity is comparable to the global maximumvorticity. Further, we denote L(t) as the arclength of L t , n theunit normal vector of L t , and κ the curvature of L t .

Theorem 1 (Deng–Hou–Yu [8], 2005). Assume that (1)

maxL t (|u · ξ | + |u · n|) ≤ CU (T − t)−A with A < 1,and (2) CL(T − t)B

≤ L(t) ≤ C0/ maxL t (|κ|, |∇ · ξ |) for0 ≤ t < T . Then the solution of the 3D Euler equations remainsregular up to t = T if A + B < 1.

In Kerr’s computations, the first condition of Theorem 1 issatisfied with A = 1/2 if we use ‖u‖∞ ≤ C(T − t)−1/2 asalleged in [18]. Kerr’s computations suggested that κ and ∇ · ξ

are bounded by O((T − t)−1/2) in the inner region of size(T − t)1/2

× (T − t)1/2× (T − t) [18]. Moreover, the length of

the vortex tube in the inner region is of order (T − t)1/2. If wechoose a vortex line segment of length (T −t)1/2 (i.e. B = 1/2),then the second condition is satisfied. However, we violate thecondition A + B < 1. Thus Kerr’s computations fall intothe critical case of Theorem 1. In a subsequent paper [9],Deng–Hou–Yu improved the non-blowup condition to includethe critical case, A + B = 1.

Theorem 2 (Deng–Hou–Yu [9], 2006). Under the sameassumptions as Theorem 1, in the case of A + B = 1, thesolution of the 3D Euler equations remains regular up to t = Tif the scaling constants CU , CL and C0 satisfy an algebraicinequality, f (CU , CL , C0) > 0.

We remark that this algebraic inequality can be checkednumerically if we obtain a good estimate of these scalingconstants. For example, if C0 = 0.1, which seems reasonablesince the vortex lines are relatively straight in the inner region,Theorem 2 would imply no blowup up to T if 2CU < 0.43CL .Unfortunately, there was no estimate available for these scalingconstants in [17]. One of our original motivations to repeatKerr’s computations using higher resolutions was to obtain agood estimate for these scaling constants.

4. The high resolution 3D Euler computations of Hou andLi [14,15]

In [14,15], we repeat Kerr’s computations using two pseudo-spectral methods. The first pseudo-spectral method uses thestandard 2/3 dealiasing rule to remove the aliasing error.For the second pseudo-spectral method, we use a novel 36thorder Fourier smoothing to remove the aliasing error. For theFourier smoothing method, we use a Fourier smoother along thex j direction as follows: ρ(2k j/N j ) ≡ exp(−36(2k j/N j )

36),where k j is the wave number (|k j | ≤ N j/2). The timeintegration is performed by using the classical fourth orderRunge–Kutta scheme. Adaptive time stepping is used to satisfythe CFL stability condition with CFL number equal to π/4. Inorder to perform a careful resolution study, we use a sequenceof resolutions: 768 × 512 × 1536, 1024 × 768 × 2048 and1536 × 1024 × 3072 in our computations. We compute thesolution up to t = 19, beyond the alleged singularity time T =

18.7 by Kerr [18]. Our computations were performed using

T.Y. Hou, R. Li / Physica D 237 (2008) 1937–1944 1939

256 parallel processors with maximal memory consumption120 Gb. The largest number of grid points is close to 5 billions.

As a first step, we demonstrate that the two pseudo-spectralmethods can be used to compute a singular solution arbitrarilyclose to the singularity time. For this purpose, we perform acareful convergence study of the two pseudo-spectral methodsin both physical and spectral spaces for the 1D inviscid Burgersequation. The advantage of using the inviscid 1D Burgersequation is that it shares some essential difficulties as the 3DEuler equations, yet we have a semi-analytic formulation forits solution. By using the Newton iterative method, we canobtain an approximate solution to the exact solution up to 13digits of accuracy. Moreover, we know exactly when a shocksingularity will form in time. This enables us to perform acareful convergence study in both the physical space and thespectral space very close to the singularity time.

We have performed a sequence of resolution study withthe largest resolution being N = 16, 384 [15]. Our extensivenumerical results demonstrate that the pseudo-spectral methodwith the high order Fourier smoothing (the Fourier smoothingmethod for short) gives a much more accurate approximationthan the pseudo-spectral method with the 2/3 dealiasing rule(the 2/3 dealiasing method for short). One of the interestingobservations is that the unfiltered high frequency coefficientsin the Fourier smoothing method approximate accurately thecorresponding exact Fourier coefficients. Moreover, we observethat the Fourier smoothing method captures about 12 ∼

15% more effective Fourier modes than the 2/3 dealiasingmethod in each dimension, see Fig. 1. The gain is even higherfor the 3D Euler equations since the number of effectivemodes in the Fourier smoothing method is higher in threedimensions. Further, we find that the error produced by theFourier smoothing method is highly localized near the regionwhere the solution is most singular. In fact, the pointwise errordecays exponentially fast away from the location of the shocksingularities. On the other hand, the error produced by the2/3 dealiasing method spreads out to the entire domain as weapproach the singularity time, see Fig. 2.

Next, we present our high resolution computations for thetwo anti-parallel vortex tubes [14]. We used the same initialcondition whose analytic formula was given by Kerr (seeSection III of [17], and also [14] for corrections of some typosin the description of the initial condition in [17]). However,there is some difference between our discretization andKerr’s discretization. We used a pseudo-spectral discretizationin all three directions, while Kerr used a pseudo-spectraldiscretization only in the x and y directions and used aChebyshev discretization in the z direction. Based on the resultsof early tests, positive vorticity in the symmetry plane wasimposed in the initial condition of Kerr [17]. How this wasimposed as the vorticity field was mapped onto the Chebyshevmesh was not documented by Kerr [17]. This has led to someambiguity in reproducing that initial condition which is beingresolved by Kerr’s group (private communication).

We first illustrate the dynamic evolution of the vortex tubes.In Figs. 4 and 5, we plot the isosurface of the 3D vortex tubes att = 0 and t = 6 respectively. As we can see, the two initial

Fig. 1. Spectra comparison on different resolutions at a sequence of moments.The additional modes that kept the Fourier smoothing method higher than the2/3rd dealiasing method are in fact correct. The initial condition is u0(x) =

sin(x). The singularity time for this initial condition is T = 1.

Fig. 2. Pointwise errors of the two pseudo-spectral methods as functions oftime using different resolutions. The plot is in a log scale. The error of the 2/3rddealiasing method (the top curve) is highly oscillatory and spreads out over theentire domain, while the error of the Fourier smoothing method (the bottomcurve) is highly localized near the location of the shock singularity.

vortex tubes are very smooth and relatively symmetric. Dueto the mutual attraction of the two anti-parallel vortex tubes,the two vortex tubes approach each other and become flatteneddynamically. By time t = 6, there is already a significantflattening near the center of the tubes. In Fig. 6, we plot thelocal 3D vortex structure of the upper vortex tube at t = 17.By this time, the 3D vortex tube has essentially turned into athin vortex sheet with rapidly decreasing thickness. The vortexlines become relatively straight. The vortex sheet rolls up nearthe left edge of the sheet.

We would like to make a few important observations. First ofall, the maximum vorticity at later stage of the computation isactually located near the rolled-up region of the vortex sheet

1940 T.Y. Hou, R. Li / Physica D 237 (2008) 1937–1944

Fig. 3. The energy spectra vs wave numbers. The dashed lines and dash-dottedlines are the energy spectra with the resolution 1024 × 768 × 2048 using the2/3 dealiasing rule and the Fourier smoothing, respectively. The times for thespectra lines are at t = 15, 16, 17, 18, 19 respectively.

Fig. 4. The 3D view of the vortex tube at t = 0.

and moves away from the bottom of the vortex sheet. Thusthe mechanism of strong compression between the two vortextubes becomes weaker dynamically at later time. Secondly, thelocation of maximum strain and that of maximum vorticityseparate as time increases. Thirdly, the relatively “strong”growth of the maximum velocity between t = 15 and t = 17becomes saturated after t = 17 when the location of maximumvorticity moves to the rolled-up region, see Fig. 7. All thesefactors contribute to the dynamic depletion of vortex stretching.The origin of this behavior need to be analyzed in the futurestudy.

We have performed a convergence study for the twonumerical methods using a sequence of resolutions. For theFourier smoothing method, we use the resolutions 768 × 512 ×

1536, 1024×768×2048, and 1536×1024×3072 respectively.Except for the computation on the largest resolution 1536 ×

1024×3072, all computations are carried out from t = 0 to t =

19. The computation on the final resolution 1536×1024×3072is started from t = 10 with the initial condition given by thecomputation with the resolution 1024×768×2048. For the 2/3dealiasing method, we use the resolutions 512 × 384 × 1024,

Fig. 5. The 3D view of the vortex tube at t = 6.

Fig. 6. The local 3D vortex structures and vortex lines around the maximumvorticity at t = 17.

Fig. 7. Maximum velocity ‖u‖∞ in time using three different resolutions.

768 × 512 × 1536 and 1024 × 768 × 2048 respectively. Thecomputations using these three resolutions are all carried outfrom t = 0 to t = 19. See [14,15] for more details.

In Fig. 3, we compare the Fourier spectra of the energyobtained by using the 2/3 dealiasing method with thoseobtained by the Fourier smoothing method. For a fixedresolution 1024×768×2048, we can see that the Fourier spectraobtained by the Fourier smoothing method retain more effective

T.Y. Hou, R. Li / Physica D 237 (2008) 1937–1944 1941

Fourier modes than those obtained by the 2/3 dealiasingmethod. This can be seen by comparing the results with thecorresponding computations using a higher resolution 1536 ×

1024 × 3072 (the solid lines). Moreover, the Fourier smoothingmethod does not give the spurious oscillations in the Fourierspectra. In comparison, the Fourier spectra obtained by the 2/3dealiasing method produce some spurious oscillations near the2/3 cut-off point. We would like to emphasize that our Fouriersmoothing method conserves the total energy extremely well,at least up to six digits of accuracy. More studies including theconvergence of the enstrophy spectra can be found in [14,15].

It is worth emphasizing that a significant portion of thoseFourier modes beyond the 2/3 cut-off position are still accuratefor the Fourier smoothing method. This portion of the Fouriermodes that go beyond the 2/3 cut-off point is about 12 ∼

15% of total number of modes in each dimension. For 3Dproblems, the total number of effective modes in the Fouriersmoothing method is about 20% more than that in the 2/3dealiasing method. For our largest resolution, we have about4.8 billions unknowns. An increase of 20% effective Fouriermodes represents a very significant increase in the resolutionfor a large scale computation.

5. Dynamics depletion of vortex stretching

In this section, we present some convincing numericalevidences which show that there is a strong dynamic depletionof vortex stretching due to local geometric regularity of thevortex lines. We first present the result on the growth of themaximum velocity in time, see Fig. 7. The growth rate ofthe maximum velocity plays a critical role in the non-blowupcriteria of Deng–Hou–Yu [8,9]. As we can see from Fig. 7,the maximum velocity remains bounded up to t = 19. Thisis in contrast with the claim in [18] that the maximum velocityblows up like O((T − t)−1/2) with T = 18.7. We note thatthe velocity field is smoother than the vorticity field. Thus itis easier to resolve the velocity field than the vorticity field.We observe an excellent agreement between the maximumvelocity fields computed by the two largest resolutions. Sincethe velocity field is bounded, the first condition of Theorem 1is satisfied by taking A = 0. Furthermore, since both ∇ · ξ andκ are bounded by O((T − t)−1/2) in the inner region of size(T − t)1/2

× (T − t)1/2× (T − t) [18], the second condition of

Theorem 1 is satisfied with B = 1/2 by taking a segment of thevortex line with length (T − t)1/2 within this inner region. ThusTheorem 1 can be applied to our computation, which impliesthat the solution of the 3D Euler equations remains smooth atleast up to T = 19.

We also study the maximum vorticity as a function of time.The maximum vorticity is found to increase rapidly from theinitial value of 0.669 to 23.46 at the final time t = 19, a factorof 35 increase from its initial value. Our computations showno sign of finite time blowup of the 3D Euler equations up toT = 19, beyond the singularity time predicted by Kerr. Themaximum vorticity computed by resolution 1024 × 768 × 2048agrees very well with that computed by resolution 1536 ×

1024 × 3072 up to t = 17.5. There is some mild disagreement

Fig. 8. Study of the vortex stretching term in time, resolution 1536 × 1024 ×

3072. The fact |ξ · ∇u · ω| ≤ c1|ω| log |ω| plus DDt |ω| = ξ · ∇u · ω implies |ω|

bounded by doubly exponential..

Fig. 9. The plot of log log ‖ω‖∞ vs time, resolution 1536 × 1024 × 3072.

toward the end of the computation. This indicates that a veryhigh space resolution is needed to capture the rapid growth ofmaximum vorticity at the final stage of the computation.

In order to understand the nature of the dynamic growth invorticity, we examine the degree of nonlinearity in the vortexstretching term. In Fig. 8, we plot the quantity, ‖ξ · ∇u · ω‖∞,as a function of time. If the maximum vorticity indeed blew uplike O((T − t)−1), as alleged in [17], this quantity should havebeen quadratic as a function of maximum vorticity. We find thatthere is tremendous cancellation in this vortex stretching term.It actually grows slower than C‖Eω‖∞ log(‖Eω‖∞), see Fig. 8. Itis easy to show that ‖ξ ·∇u ·ω‖∞ ≤ C‖Eω‖∞ log(‖Eω‖∞) wouldimply at most doubly exponential growth in the maximumvorticity. Indeed, as demonstrated by Fig. 9, the maximumvorticity does not grow faster than doubly exponential in time.We have also generated the similar plot by extracting the data

1942 T.Y. Hou, R. Li / Physica D 237 (2008) 1937–1944

Fig. 10. The energy spectra for velocity at t = 15, 16, 17, 18, 19 (from bottomto top) in log–log scale. The dashed line corresponds to k−3.

from Kerr’s paper [17]. We find that log(log(‖ω‖∞)) basicallyscales linearly with respect to t from 14 ≤ t ≤ 17.5 whenhis computations are still reasonably resolved. This implies thatthe maximum vorticity up to t = 17.5 in Kerr’s computationsdoes not grow faster than doubly exponential in time. This isconsistent with our conclusion.

We study the decay rate in the energy spectrum in Fig. 10at t = 16, 17, 18, 19. A finite time blowup of enstrophy wouldimply that the energy spectrum decays no faster than |k|

−3. Ourcomputations show that the energy spectrum approaches |k|

−3

for |k| ≤ 100 as time increases to t = 19. This is in qualitativeagreement with Kerr’s results. Note that there are only less than100 modes available along the |kx | or |ky | direction in Kerr’scomputations, see Fig. 18 (a)–(b) of [17]. On the other hand, ourcomputations show that the high frequency Fourier spectrumfor 100 ≤ |k| ≤ 1300 decays much faster than |k|

−3, as onecan see from Fig. 10. This indicates that there is no blowup inenstrophy.

It is interesting to ask how the vorticity vector aligns with theeigenvectors of the deformation tensor. Recall that the vorticityequations can be written as [20]

∂

∂tω + (u · ∇)ω = S · ω, S =

12(∇u + ∇

T u). (1)

Let λ1 < λ2 < λ3 be the three eigenvalues of S. Theincompressibility condition implies that λ1 + λ2 + λ3 = 0. Ifthe vorticity vector aligns with the eigenvector corresponding toλ3, which gives the maximum rate of stretching, then it is verylikely that the 3D Euler equations would blow up in a finitetime.

In Table 1, we document the alignment information of thevorticity vector around the point of maximum vorticity withresolution 1536 × 1024 × 3072. In this table, θi is the anglebetween the i-th eigenvector of S and the vorticity vector. Onecan see clearly that for 16 ≤ t ≤ 19 the vorticity vector at thepoint of maximum vorticity is almost perfectly aligned with thesecond eigenvector of S. Note that the second eigenvalue, λ2,

Table 1The alignment of the vorticity vector and the eigenvectors of S around the pointof maximum vorticity with resolution 1536 × 1024 × 3072

Time |ω| λ1 θ1 λ2 θ2 λ3 θ3

16.012 5.628 −1.508 89.992 0.206 0.007 1.302 89.99816.515 7.016 −1.864 89.995 0.232 0.010 1.631 89.99017.013 8.910 −2.322 89.998 0.254 0.006 2.066 89.99317.515 11.430 −2.630 89.969 0.224 0.085 2.415 89.92018.011 14.890 −3.625 89.969 0.257 0.036 3.378 89.97918.516 19.130 −4.501 89.966 0.246 0.036 4.274 89.98419.014 23.590 −5.477 89.966 0.247 0.034 5.258 89.994

Here, θi is the angle between the i th eigenvector of S and the vorticity vector.

is positive and is about 20 times smaller in magnitude than thelargest and the smallest eigenvalues. Although the alignmentof the vorticity vector with the second eigenvector of thedeformation tensor does not rule out a finite time blowup, thisalignment is another indication that there is a strong dynamicdepletion of vortex stretching.

6. The Kida–Pelz high-symmetry data

Another well-known numerical evidence for finite timeEuler singularities is the Kida–Pelz high-symmetry initial data[3,19]. Some people have argued that the singular solution ofthe 3D Euler equations, if it exists, could be very unstable. Ahighly symmetric initial condition may have a better chanceto produce a finite time singularity. It is also believed thata computer code needs to build in this symmetry propertyexplicitly in order to capture the potentially unstable singularsolution. This consideration motivated Boratav and Pelz toperform numerical simulations using a high-symmetry initialcondition for the Navier–Stokes equations in [3].

The initial condition that Boratav and Pelz used [3] hasthe rotational symmetry and the permutation symmetry, whichwas first introduced by Kida [19]. Their simulations suggesteda possible finite time blowup of the maximum vorticity inthe limit of infinite Reynolds numbers. However, as theyrealized later, their simulations were under-resolved at latertimes when the solution became nearly singular. The vortexstructure near the region of maximum vorticity motivated Pelzto construct a vortex filament model to understand this singularbehavior. In [21], Pelz presented some numerical evidenceswhich suggest that his filament model develop a self-similarblowup in a finite time. It is interesting to note that Pelz’sself singular solution also falls into the critical case of theDeng–Hou–Yu local non-blowup criteria (see Theorem 2). Tounderstand if the same initial condition that led to a finite timeblowup in Pelz’s filament model would lead to a finite timeblowup of the full 3D Euler equations, we decide to repeatPelz’s computations.

Pelz’s original filament model was designed for the entirefree space. To perform the numerical simulation of the 3DEuler equations in the free space R3 is very expensive. Asa first step, we derive a corresponding periodic filamentmodel. The periodic filament model involves an infinite sumover all the periodic images of the Biot–Savart kernel. This

T.Y. Hou, R. Li / Physica D 237 (2008) 1937–1944 1943

Fig. 11. The validity check of singularity fitting using the asymptoticexpression ‖u‖∞ =

C√tcrit−t . The figure shows tcrit as a function of the

computational steps, with tcrit → 0.0257874. Adaptive time stepping is usedwith the time step chosen to be proportional to the inverse of ‖u‖∞.

Fig. 12. The locations of the filaments at the end of our computation. The figuregives a closeup view of the filaments around the origin.

makes the computation of the periodic filament kernel moreexpensive than the one over the free space. To reduce thecomputational cost, we apply the Ewald summation formula,which significantly reduces the computational cost.

We solve the periodic filament model using an initialcondition which is qualitatively the same as the one usedby Pelz [21]. Our numerical computations show that theperiodic filament model indeed develops a finite time self-similar singularity around t = 0.0257874, see Figs. 11 and 12.However, when we use the same initial condition to solve thefull 3D Euler equations, we find that the solution of the 3DEuler equations has a completely different behavior from thatof the filament model. We observe no finite time singularityfor the 3D Euler equations using the same initial condition.We use a sequence of space resolutions with the two largestresolutions being 10243 and 20483. More than 100Gb memoryis used in our computation on the 20483 computations. As wecan see from Figs. 13 and 14, the growth of maximum vorticity

Fig. 13. Maximum vorticity in time of the full Euler equations with tworesolutions: 10243 (dashed line) vs N = 20484 (solid line).

Fig. 14. Maximum velocity in time of the full Euler equations with resolution:10243. The maximum velocity seems to saturate at a later time.

in time is very mild. The maximum velocity is bounded andbecomes saturated around t = 0.0325. The 3D isosurface of thevortex tubes at t = 0.03 plotted in Fig. 15 also shows that thevortex tubes remain quite regular. We remark that Grauer andhis coworkers have recently carried out the full Euler simulationusing a simplified Pelz’s high-symmetry initial condition whichconsists of 12 straight parallel bars [13]. They find that thevortex tubes become severely flattened as they approach eachother and the growth of maximum vorticity is only exponentialin time.

Finally, we remark that we have repeated Boratav’sand Pelz’s Navier–Stokes computations [3] using the sameinitial condition, building both the rotational and permutationsymmetries of the solution explicitly into our code. Ourresolution study shows that their computations are resolvedonly up to t = 1.6 when the growth of the maximum vorticity

1944 T.Y. Hou, R. Li / Physica D 237 (2008) 1937–1944

Fig. 15. The 50% isosurface of | Eω| at t = 0.03. Full 3D Euler equations.

is only exponential in time. The nearly singular growth ofmaximum vorticity around t = 2.06 seems due to under-resolution.

7. Concluding remarks

Our analysis and computations reveal a subtle dynamicdepletion of vortex stretching. Sufficient numerical resolution isessential in capturing this dynamic depletion. Our computationsfor the two anti-parallel vortex tubes’ initial data and the high-symmetry initial data show that the velocity is bounded and thatthe vortex stretching term is bounded by C‖ω‖L∞ log(‖ω‖L∞).It is natural to ask if is this dynamic depletion generic? and whatis the driving mechanism for this depletion of vortex stretching?Some exciting progress has been made recently in analyzing thedynamic depletion of vortex stretching and nonlinear stabilityfor 3D axisymmetric flows with swirl [16]. The local geometricstructure of the solution near the region of maximum vorticityand the anisotropic scaling of the support of maximum vorticityseem to play a key role in the dynamic depletion of vortexstretching.

Acknowledgments

We would like to thank Prof. Lin-Bo Zhang from theInstitute of Computational Mathematics and the Center of HighPerformance Computing in Chinese Academy of Sciences forproviding us with the computing resource to perform this largescale computational project. We also thank Prof. Robert Kerrfor providing us with his Fortran subroutine that generatesinitial data. This work was supported in part by NSF under theNSF grants, FRG DMS-0353838, ITR ACI-0204932 and DMS-

0713670. Li was subsidized by the National Basic ResearchProgram of China under the grant 2005CB321701.

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